Special relativity does not seem to specifically exclude
tachyons, so long as they do not cross the lightspeed barrier and
do not interact with other particles to cause causality
violations. Quantum mechanical analyses of tachyons indicate that
even though they travel faster than light they would not be able
to carry information faster than light, thus failing to violate
causality. But in this case, if tachyons are by their very nature
indetectable, it brings into question how real they might be.

The argument demonstrating that tachyons (should they
exist, of course) cannot carry an electric charge. For a
(imaginary-massed) particle travelling faster than c, the less
energy the tachyon has, the faster it travels, until at zero
energy the tachyon is travelling with infinite velocity, or is
transcendent. Now a charged tachyon at a given (non-infinite)
speed will be travelling faster than light in its own medium, and
should emit Cherenkov radiation. The loss of this energy will
naturally reduce the energy of the tachyon, which will make it go
faster, resulting in a runaway reaction where any charged tachyon
will promptly race off to transcendence.

Although the above argument results in a curious conclusion,
the meat of the tachyon paradox is this: In relativity, the
transcendence of a tachyon is frame-dependent. That is, while a
tachyon might appear to be transcendent in one frame, it would
appear to others to still have a nonzero energy. But in this case
we have a situation where in one frame it would have come to zero
energy and would stop emitting Cherenov radiation, but in another
frame it would still have energy left and should be emitting
Cherenkov radiation on its way to transcendence. Since they
cannot both be true, by relativistic arguments, tachyons cannot be
charged.

This argument naturally does not make any account of quantum
mechanical treatments of tachyons, which complicate the situation
a great deal.

Satellites which orbit a body at one or the other Trojan points
relative to a secondary body. There are several examples of this
in our own solar system: a group of asteroids which orbit in the
the Trojan points of Jupiter; daughter satellites which orbit in
the Trojan points of the Saturn-Tethys system, and an additional
satellite (Helene) which orbits in the forward Trojan point
of Saturn and Dione.

That is actually not the paradox. The paradox stems from
attempting to naively analyze the situation to figure out why.
From Henrik's point of view (and from everyone else on Earth),
Albert seems to speed off for a long time, linger around, and then
return. Thus he should be the younger one, which is what we see.
But from Albert's point of view, it's Henrik (and the whole of the
Earth) that are travelling, not he. According to special
relativity, if Henrik is moving relative to Albert, then Albert
should measure his clock as ticking slower -- and thus Henrik is
the one who should be younger. But this is not what happens.

So what's wrong with our analysis? The key point here is that
the symmetry was broken. Albert did something that Henrik did
not -- Albert accelerated in turning around. Henrik did no
accelerating, as he and all the other people on the Earth can
attest to (neglecting gravity). So Albert broke the symmetry, and
when he returns, he is the younger one.